extensive game with perfect information - ucla econ · example 2: stackelberg competion consider...

Extensive Game with Perfect Information

Ichiro Obara

UCLA

February 20, 2012

Obara (UCLA) Extensive Game with Perfect Information February 20, 2012 1 / 14

Extensive Game with Perfect Information

Extensive Game with Perfect Information

We study dynamic games where players make a choice sequentially.

We assume perfect information: each player can perfectly observe

the past actions.

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Extensive Game with Perfect Information

Example 1: Chain Store Game

A chain store (CS) has a branch in a city

There is one potential competitor (C) in the city.

The game proceeds a follows:

I C decides whether to enter the market or not.

I Given C’s choice, CS decides whether to accommodate or fight back.

The profits are (0, 0) (CS’s profit, C’s profit) if C enters and CS fights

back, (2, 2) if C enters and CS accommodates, and (5, 1) if C does

not enter.

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Extensive Game with Perfect Information

Example 1: Chain Store Game

This game can be described as follows.

Out

IN

A

F 0, 0

2, 2

5, 1

C

CS

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Extensive Game with Perfect Information

Example 2: Stackelberg Competion

Consider the environment of the standard Cournot duopoly model.

Suppose that the firms make decision sequentially.

I Firm 1 (leader) first chooses how much to produce.

I Then firm 2 (follower) decides how much to produce.

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Extensive Game with Perfect Information

Example 2: Stackelberg Competion

This game looks like

q1

q2

p1(q1, q2),

p2(q1, q2)

1

2

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Extensive Game with Perfect Information

Formal Model

Extensive Game with Perfect Information

Extensive game with Perfect Information consists of

I a finite set N

I a set of sequences H such that

F ∅ ∈ H

F (a1, ...., ak) ∈ H → (a1, ...., a`) ∈ H for any ` < k

F (a1, ....) ∈ H if (a1, ...., ak) ∈ H for k = 1, 2, ....

with Z ⊂ H defined by (a1, ...., ak) ∈ Z ⇔6 ∃ak+1, (a1, ...., ak+1) ∈ H.

I a function P : H/Z → N

I a function Vi : Z → < for i ∈ N.

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Extensive Game with Perfect Information

N is the set of players.

H is the set of histories with

I Z as the set of terminal histories, and

I ∅ as the initial history.

P specifies who makes a choice at each history.

Vi (z) is player i ’s payoff at terminal history z .

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Extensive Game with Perfect Information

Let Hi ⊂ H be the subset of histories such that P(h) = i . This is the

set of histories where player i makes a choice.

At history h ∈ H/Z , the set of actions that are available to player

P(h) is

A(h) = {a|(h, a) ∈ H}

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Extensive Game with Perfect Information

Strategy

Player i ’s strategy for extensive game (N,H,P, (Vi )) is a mapping si

that assigns an action in A(h) at each h ∈ Hi . Let Si be the set of

player i ’s strategies.

Every strategy profile s = (s1, ..., sn) defines an outcome

O(s) = (a1, ..., aK ) ∈ Z (K may be ∞) by

I sP(∅)(∅) = a1

I sP(a1)(a1) = a2

I sP(a1,a2)(a1, a2) = a3....

Thus player i ’s payoff is Vi (O(s)) given a strategy profile s.

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Extensive Game with Perfect Information

Strategy

A strategy is not just a contingent plan of actions. It specifies an action at every

history, even at histories that are never reached given the strategy(ex. strategy Ba

for the game below).

1

2

1

A B

C D

a b

0,1 -1,-2

0,0

2,0

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Extensive Game with Perfect Information

Mixed Strategy and Behavior Strategy

As in strategic games, we can define a mixed strategy for extensive games as

a probability distribution over pure strategies (∆(Si )).

There is another way to express a mixed strategy. Player i ’s behavioral

strategy σi is a mapping from Hi to a distribution on the set of available

actions (σi (h) ∈ ∆(Ai (h)) for each h ∈ H).

They are different representations of the same thing. Every behavior

strategy is clearly a mixed strategy. Every mixed strategy can be replicated

by a behavior strategy.

We will use behavior strategy representation most of the time.

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Extensive Game with Perfect Information

Nash Equilibrium

Note that an extensive game with perfect information (N,H,P, (Vi ))

determines a strategic game (N, (Si ), (Vi )). So we can define Nash

equilibrium for extensive game with perfect information.

Nash Equilibrium

For extensive game with perfect information (N,H,P, (Vi )), a profile of

strategies s∗ is a Nash equilibrium if

Vi (O(s∗)) ≥ Vi (O(s ′i , s∗−i ))

for any s ′i ∈ Si and any i ∈ N.

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Extensive Game with Perfect Information

Nash equilibrium is often too permissive.

For the chain store game, there exists two NE: (In,A) and (Out,F ).

One may argue that (Out,F ) is less reasonable, because F is not an

optimal action once “In” is chosen.

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